Designing the program trajectory for steering a spacecraft under arbitrary boundary conditions

Abstract

A problem is considered of designing the program trajectory of a spacecraft turning from an arbitrary initial orientation to an arbitrary final orientation, with the orientations being defined with unit quaternions. A projection of a group of unit quaternions Sp(1) on a sphere with the radius of 2ˇ is used to represent rotation of a body as a motion of a point inside the given sphere. Polynomials of the fifth degree are considered as a class of functions to define the program trajectories in the sphere. The suggested class of trajectories is demonstrated to be effective to provide the possibility of meeting boundary conditions at arbitrary values of velocities and accelerations.

On One Method for Constructing a Programmed Trajectory in Problems of Stabilizing a Spacecraft

“…As noted in [5], the set of unit quaternions defining the orientation can be associated with the points of the ball with the radius π of the three-dimensional space, taking this into account the problem of finding the program trajectory q( ) t can be replaced by the problem of finding its three-dimensional image ( ) t r in the ball with the radius π . Connection between the coordinates of the points of the ball and components of the quaternion was established in [6]:…”